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About Logic and Logicians About Logic and Logicians A palimpsest of essays by Georg Kreisel Selected and arranged by Piergiorgio Odifreddi Volume 1: Philosophy Lógica no Avião ABOUT LOGIC AND LOGICIANS A palimpsest of essays by Georg KREISEL Selected and arranged by Piergiorgio ODIFREDDI Volume I. Philosophy. Editorial Board Fernando Ferreira Departamento de Matem´atica Universidade de Lisboa Francisco Miraglia Departamento de Matem´atica Universidade de S~aoPaulo Graham Priest Department of Philosophy The City University of New York Johan van Benthem Department of Philosophy Stanford University Tsinghua University University of Amsterdam Matteo Viale Dipartimento di Matematica \Giuseppe Peano" Universit`adi Torino Piergiorgio Odifreddi, About Logic and Logicians, Volume 1. Bras´ılia: L´ogicano Avi~ao,2019. S´erieA, Volume 1 I.S.B.N. 978-65-900390-2-6 Prefixo Editorial 900390 Obra publicada com o apoio do PPGFIL/UnB. Editor's Preface These books are a first version of Odifreddi's collection of Kreisel's expository papers, which together constitute an extensive, scholarly account of the philo- sophical and mathematical development of many of the most important figures of modern logic; some of those papers are published here for the first time. Odifreddi and Kreisel worked together on these books for several years, and they are the product of long discussions. They finally decided that they would collect those essays of a more expository nature, such as the biographical memoirs of the fellows of the Royal Society (of which Kreisel himself was a member) and other related works. Also included are lecture notes that Kreisel distributed in his classes, such as the first essay printed here, which is on the philosophy of mathematics and geometry. Kreisel himself wrote all the texts, but Odifreddi has made some substantial editorial interventions, rearranging some of the material, breaking the text into sections and paragraphs, inserting titles, moving or removing some notes, and eliminating some digressions. These interventions were made in order to give the essays some of their original freshness and linearity, qualities that were lost in later versions. Some other minor modifications were made here and there, consisting basi- cally in the correction of a small number of erroneous references in the original manuscripts. Kreisel's expository works are invaluable to logicians, and we hope that the reader may find the present edition to his advantage, even if there is still some editorial work to do. Rodrigo Freire. Bras´ılia,April 2019. Contents I INTRODUCTION 1 1 INTRODUCTION TO THE PHILOSOPHY OF MATHEMA- TICS 2 Introduction . .3 Warnings . .4 1.1 Definitions ..............................4 Philosophical perplexities (putting one's questions into question) .5 Heuristic value of perplexities: a source of issues and distinctions .6 Examples of distinctions (to be developed later) . .8 Warnings: what not to expect of our elementary (i.e. general) dis- tinctions . .9 A new philosophical problem . 10 1.2 The Circle: Two Aspects of Explicit Definitions ...... 11 A first definition of circle . 11 A second definition of circle . 13 A philosophical scheme and the choice between the two definitions of the circle . 14 Popular philosophical literature . 18 Definitions as an auxiliary means . 19 Eliminating definitions (from proofs) . 21 On explicit definitions . 23 1.3 Area: Two Aspects of Implicit Definitions .......... 24 The area of a rectangle . 24 Equality of area . 26 Surprises, perplexities and (promising) problems . 27 Measure of area . 31 Grand philosophical questions . 34 II ON ARISTOTLE 36 2 THE LITERARY VALUE OF ARISTOTLE'S THOUGHT 37 The value of great philosophers . 37 CONTENTS Analogies . 39 Abstractions (also called structures) . 39 Choice of abstractions . 40 What we know and how we know . 42 Disclaimer . 43 3 ARISTOTLE'S LOGIC TODAY 44 3.1 Obsolete Ideas ........................... 44 The infinite . 44 Logic . 45 3.2 Less Obsolete Ideas ........................ 46 Particular proofs and general proofs . 46 Propositional Logic . 48 3.3 Eternal Verities ........................... 50 Exact philosophy . 51 III ON FREGE 53 4 FREGE'S FOUNDATIONS 54 4.1 Background: Some Discoveries ................. 55 Functions . 55 Isomorphism . 56 The structure of the natural numbers . 56 Defining functions by equations . 57 Order . 58 The structure of the rational numbers . 58 The structure of the real numbers . 59 4.2 Set Theory .............................. 59 Set-theoretic purity . 59 Supply of sets . 60 Defining sets by predicates . 60 Proofs in Set Theory . 61 4.3 Formal Proofs ............................ 62 Formal Procedures . 62 Formal procedures in logic . 63 Formal procedures in mathematics . 64 4.4 After Frege's Foundations .................... 64 Newton's foundations . 65 Bourbaki's foundations . 66 Philosophical perspective . 68 Some neglected concerns . 70 CONTENTS 5 VARIANTS OF FREGE FOUNDATIONS 72 5.1 Zermelo's foundations ....................... 73 5.2 Hilbert's foundations ....................... 74 5.3 Brouwer's foundations ...................... 76 Propositions about incompletely defined objects . 76 Parallels between the theories of sets and choice sequences . 78 Proofs and logical operations applied to propositions about incom- pletely defined objects . 80 5.4 Philosophical Hygiene ....................... 82 5.5 Bourbaki's alternative ....................... 84 Warnings to literal-minded readers . 85 IV ON RUSSELL 86 6 RUSSELL'S LOGIC 87 Logic . 88 The theory of descriptions . 89 Russell's paradox . 89 The doctrine of types . 91 Principia Mathematica . 91 7 PRINCIPIA MATHEMATICA: CRITICAL OR SPECULATIVE PHILOSOPHY? 92 Mechanization of information . 93 7.1 Critical Philosophy ........................ 95 Antinomies . 97 7.2 Speculative Philosophy ...................... 99 What we know versus how we know . 101 7.3 Looking Back. Russell's Hopes and Disappointments ... 103 8 BERTRAND RUSSELL 107 8.1 Russell's Life ............................ 108 Childhood . 108 Adolescence . 109 Studies and research . 109 Wittgenstein . 110 First world war . 111 Between the wars . 112 Return to England . 114 8.2 Mathematical Logic and Logical Foundations of Mathe- matics ................................. 116 Foundations: loaded terminology . 116 CONTENTS Background: logical language . 117 Background: sets and predicates . 119 Background: definitions of natural and real numbers . 121 New theories: Russell's paradox . 122 True, false, meaningless . 124 Doctrine of types and use of types . 126 Principia Mathematica and the axiomatization of mathematical practice . 129 Principia Mathematica: a parenthesis in the refutation of Kant . 132 8.3 Philosophy, Pedagogy, Literature ................ 133 Scope of philosophy . 133 Uncertainty and generality . 134 Critical philosophy and Occam's razor . 136 From what we know to how we know . 138 Pedagogy in the large . 140 Pedagogy in the small . 141 Philosophic contemplation . 143 Russell's picture of America . 144 V ON WITTGENSTEIN 147 9 ON SOME CONVERSATIONS WITH WITTGENSTEIN 148 Constructive content of proofs . 149 Family resemblances of concepts . 151 `Deep' processes . 153 Warning . 155 10 LUDWIG WITTGENSTEIN 157 Meaning of `philosophy': adaptation to background knowledge . 157 Style of the chapter . 160 10.1 Wittgenstein: First and Second Thoughts .......... 160 A universal language for definitions . 161 Other functions of language . 163 Manifesto . 163 10.2 Biographical Material ....................... 164 Ancestry . 165 The father . 166 (Homo)sexuality . 167 The youngest sister . 168 The rest of the family . 169 The house in the Kundtmanngasse . 170 Music . 170 CONTENTS 10.3 Tractatus ............................... 171 Propositional logic . 171 Finite Problems . 172 Infinite problems . 174 Language: expressive and reasoning capacities . 175 Anti-psychologism (another clumsy word for timely advice) . 176 Specific Reminders . 176 Formal rules for propositional logic . 177 Discussion . 180 10.4 Transition to Wittgenstein's Later Concerns ........ 181 Indignation . 182 Artistic skill . ..
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